Nonnormality in Lyapunov Equations

The singular values of the solution to a Lyapunov equation determine the potential accuracy of the low-rank approximations constructed by iterative methods. Low- rank solutions are more accurate if most of the singular values are small, so a priori bounds that describe coefficient matrix properties that correspond to rapid singular value decay are valuable. Previous bounds take similar forms, all of which weaken (quadratically) as the coefficient matrix departs from normality. Such bounds suggest that the more nonnormal the coefficient matrix becomes, the slower the singular values of the solution will decay. However, simple examples typically exhibit an eventual acceleration of decay if the coefficient becomes very nonnormal. We will show that this principle is universal: decay always improves as departure from normality increases beyond a given threshold, specifically as the numerical range of the coefficient matrix extends farther into the right half-plane. We also give examples showing that similar behavior can occur for general Sylvester equations, though the right-hand side plays a more important role

Network structural origin of instabilities in large complex systems

A central issue in the study of large complex network systems, such as power grids, financial networks, and ecological systems, is to understand their response to dynamical perturbations. Recent studies recognize that many real networks show nonnormality and that nonnormality can give rise to reactivity--the capacity of a linearly stable system to amplify its response to perturbations, oftentimes exciting nonlinear instabilities. Here, we identify network structural properties underlying the pervasiveness of nonnormality and reactivity in real directed networks, which we establish using the most extensive data set of such networks studied in this context to date. The identified properties are imbalances between incoming and outgoing network links and paths at each node. Based on this characterization, we develop a theory that quantitatively predicts nonnormality and reactivity and explains the observed pervasiveness. We suggest that these results can be used to design, upgrade, control, and manage networks to avoid or promote network instabilities.Comment: Includes Supplementary Material

A preconditioned MINRES method for nonsymmetric Toeplitz matrices

Circulant preconditioning for symmetric Toeplitz linear systems is well established; theoretical guarantees of fast convergence for the conjugate gradient method are descriptive of the convergence seen in computations. This has led to robust and highly efficient solvers based on use of the fast Fourier transform exactly as originally envisaged in [G. Strang, Stud. Appl. Math., 74 (1986), pp. 171--176]. For nonsymmetric systems, the lack of generally descriptive convergence theory for most iterative methods of Krylov type has provided a barrier to such a comprehensive guarantee, though several methods have been proposed and some analysis of performance with the normal equations is available. In this paper, by the simple device of reordering, we rigorously establish a circulant preconditioned short recurrence Krylov subspace iterative method of minimum residual type for nonsymmetric (and possibly highly nonnormal) Toeplitz systems. Convergence estimates similar to those in the symmetric case are established

Numerical range for random matrices

We analyze the numerical range of high-dimensional random matrices, obtaining limit results and corresponding quantitative estimates in the non-limit case. For a large class of random matrices their numerical range is shown to converge to a disc. In particular, numerical range of complex Ginibre matrix almost surely converges to the disk of radius $\sqrt{2}$. Since the spectrum of non-hermitian random matrices from the Ginibre ensemble lives asymptotically in a neighborhood of the unit disk, it follows that the outer belt of width $\sqrt{2}-1$ containing no eigenvalues can be seen as a quantification the non-normality of the complex Ginibre random matrix. We also show that the numerical range of upper triangular Gaussian matrices converges to the same disk of radius $\sqrt{2}$, while all eigenvalues are equal to zero and we prove that the operator norm of such matrices converges to $\sqrt{2e}$.Comment: 23 pages, 4 figure

GMRES convergence analysis for a convection-diffusion model problem

When GMRES [Y. Saad and M. H. Schultz, SIAM J. Sci. Statist. Comput.}, 7 (1986), pp. 856--869] is applied to streamline upwind Petrov--Galerkin (SUPG) discretized convection-diffusion problems, it typically exhibits an initial period of slow convergence followed by a faster decrease of the residual norm. Several approaches were made to understand this behavior. However, the existing analyses are solely based on the matrix of the discretized system and they do not take into account any influence of the right-hand side (determined by the boundary conditions and/or source term in the PDE). Therefore they cannot explain the length of the initial period of slow convergence which is right-hand side dependent. We concentrate on a frequently used model problem with Dirichlet boundary conditions and with a constant velocity field parallel to one of the axes. Instead of the eigendecomposition of the system matrix, which is ill conditioned, we use its orthogonal transformation into a block-diagonal matrix with nonsymmetric tridiagonal Toeplitz blocks and offer an explanation of GMRES convergence. We show how the initial period of slow convergence is related to the boundary conditions and address the question why the convergence in the second stage accelerates

GMRES convergence analysis for a convection-diffusion model problem

When GMRES [Y. Saad and M. H. Schultz, SIAM J. Sci. Statist. Comput.}, 7 (1986), pp. 856--869] is applied to streamline upwind Petrov--Galerkin (SUPG) discretized convection-diffusion problems, it typically exhibits an initial period of slow convergence followed by a faster decrease of the residual norm. Several approaches were made to understand this behavior. However, the existing analyses are solely based on the matrix of the discretized system and they do not take into account any influence of the right-hand side (determined by the boundary conditions and/or source term in the PDE). Therefore they cannot explain the length of the initial period of slow convergence which is right-hand side dependent. We concentrate on a frequently used model problem with Dirichlet boundary conditions and with a constant velocity field parallel to one of the axes. Instead of the eigendecomposition of the system matrix, which is ill conditioned, we use its orthogonal transformation into a block-diagonal matrix with nonsymmetric tridiagonal Toeplitz blocks and offer an explanation of GMRES convergence. We show how the initial period of slow convergence is related to the boundary conditions and address the question why the convergence in the second stage accelerates

Modal and nonmodal stability analysis of electrohydrodynamic flow with and without cross-flow

We report the results of a complete modal and nonmodal linear stability analysis of the electrohydrodynamic flow (EHD) for the problem of electroconvection in the strong injection region. Convective cells are formed by Coulomb force in an insulating liquid residing between two plane electrodes subject to unipolar injection. Besides pure electroconvection, we also consider the case where a cross-flow is present, generated by a streamwise pressure gradient, in the form of a laminar Poiseuille flow. The effect of charge diffusion, often neglected in previous linear stability analyses, is included in the present study and a transient growth analysis, rarely considered in EHD, is carried out. In the case without cross-flow, a non-zero charge diffusion leads to a lower linear stability threshold and thus to a more unstable low. The transient growth, though enhanced by increasing charge diffusion, remains small and hence cannot fully account for the discrepancy of the linear stability threshold between theoretical and experimental results. When a cross-flow is present, increasing the strength of the electric field in the high-$Re$ Poiseuille flow yields a more unstable flow in both modal and nonmodal stability analyses. Even though the energy analysis and the input-output analysis both indicate that the energy growth directly related to the electric field is small, the electric effect enhances the lift-up mechanism. The symmetry of channel flow with respect to the centerline is broken due to the additional electric field acting in the wall-normal direction. As a result, the centers of the streamwise rolls are shifted towards the injector electrode, and the optimal spanwise wavenumber achieving maximum transient energy growth increases with the strength of the electric field

Doctor of Philosophy

dissertationThere continue to be a large number of at-risk students who do not complete high school every year. There are a number of identifiable risk factors that can contribute to an increased likelihood of students dropping out of high school. With advances in data collection, schools are now better able to identify and track students' progress towards graduation with detection systems, called Early Warning Systems (EWS). EWS utilize data on grades, behavior referrals, and attendance gathered from school records to identify students at increased risk for dropout. Students identified by schools as "at-risk" or "off-track" can then be provided with effective interventions designed to prevent dropout. Student engagement is one variable that schools have the ability to measure and potentially increase through interventions. EWS can be used to help facilitate linking "at-risk" and "off-track" students, who potentially report low school engagement, to a school's preexisting intervention programs in order to prevent dropout. Furthermore, participation in extracurricular activities provided by the school may help make students feel more connected and engaged at school. This can be particularly important for students transitioning from middle school to high school. These transition programs, set up to help connect the incoming class with upper classmates, are a great way for students to acclimate to the high school setting. With the different programs in place within a high school, it is important that students are connected with the programs and services that are right for them to help facilitate engagement and connectedness to school. Ensuring engagement and connectedness to school can positively impact grades, attendance, and behavior, and also decrease the likelihood of dropping out. The current study aimed to confirm the model that participation in at-risk programs has a positive impact on student engagement, which in turn, positively impacts student outcomes, such as grades, attendance, and behavior. The study found that participation in at-risk programs did not necessarily improve school outcomes or student engagement; however, students within these programs who reported higher school engagement had better school outcomes